CN105138001B - A kind of quadrotor attitude control method - Google Patents
A kind of quadrotor attitude control method Download PDFInfo
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- CN105138001B CN105138001B CN201510573908.6A CN201510573908A CN105138001B CN 105138001 B CN105138001 B CN 105138001B CN 201510573908 A CN201510573908 A CN 201510573908A CN 105138001 B CN105138001 B CN 105138001B
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Abstract
A kind of quadrotor attitude control method, first by given instruction attitude angle and actual posture angular error calculation amount, then chooses sliding mode and Reaching Law, designs Sliding Mode Attitude control law；Effectively to suppress to buffet caused by sliding formwork control, using sliding mode and its first differential as the input of fuzzy controller, to control gain as the output design adaptive fuzzy sliding mode attitude control law of fuzzy controller, pass through fuzzy rule on-line tuning and control gain.The closed-loop system controlled by this method can stable regulation to attitude angle is instructed, with good robustness and control accuracy, the Project Realization for quadrotor gesture stability provides effective means.
Description
Technical field
The invention belongs to automatic control technology field, and in particular to a kind of quadrotor attitude control method, it is
Quadrotor provides a kind of new method of gesture stability.
Background technology
Quadrotor is as a kind of new rotor wing unmanned aerial vehicle, with flexible deployment, fast reserve, spot hover etc.
Application characteristic, can realize VTOL, omnirange flight and spot hover in narrow space, by carrying imaging device,
The fields such as reconnaissance and surveillance, earth observation, environmental monitoring, disaster assistance, traffic monitoring and Aerial photography are can be widely applied to, are had
There are important application value and wide application prospect.Above-mentioned application is required to high-precision automatic flight control, wherein, posture
Control is quadrotor stabilized flight the most basic control requirement.The attitude motion of quadrotor has strong coupling
Conjunction, multivariable, it is non-linear, uncertain the features such as, and easily by external interference, therefore, gesture stability turns into the pass of its flight control
Key technology and difficult point.
PID (ratio, integration and differential) controls are one of the most commonly used control methods of quadrotor gesture stability,
Have the advantages that design is simple, be easy to Project Realization, but when model parameter and operating mode change, control performance is difficult to obtain
Ensure.In existing achievement in research, quadrotor attitude control law is all based on greatly linear model and is designed, and non-thread is not considered
Property and each passage between coupling, when existing, model is uncertain and control effect is poor when there is external interference.
The content of the invention
For above-mentioned technical problem, the present invention provides a kind of quadrotor attitude control method, control of the invention
System architecture is as shown in Figure 1.It designs posture using nonlinear attitude kinetic model as controlled device using sliding-mode control
Control law, is realized to model uncertainty and the robust control of external disturbance；Effectively to suppress to buffet caused by sliding formwork control, with
Sliding mode and its input that first differential is fuzzy controller, to control gain adaptive as the output design of fuzzy controller
Fuzzy Sliding Mode Attitude control law, gain is controlled by fuzzy rule on-line tuning.The closed-loop system controlled by this method can be steady
Section set the tone to instruction attitude angle, is that the engineering of quadrotor gesture stability is real with good robustness and control accuracy
Now provide effective means.
A kind of quadrotor attitude control method, its main contents and step are：First by given instruction attitude angle
With actual posture angular error calculation amount, sliding mode and Reaching Law are then chosen, Sliding Mode Attitude control law is designed；For effectively suppression
Buffeted caused by sliding formwork control, using sliding mode and its first differential as the input of fuzzy controller, to control gain to be fuzzy
The output design adaptive fuzzy sliding mode attitude control law of controller, gain is controlled by fuzzy rule on-line tuning.It is actual to answer
In, the actual attitude angle of quadrotor is obtained by integrated navigation system measurement, and obtained control will be calculated by this method
Amount transmits to executing agency the gesture stability that quadrotor can be achieved.
A kind of quadrotor attitude control method of the present invention, it is comprised the following steps that：
Step one：Given instruction attitude angle, gives instruction pitching angle thetar, instruct yaw angle ψr, instruction roll angle φr；
Step 2：The margin of error is calculated, the margin of error e between computations attitude angle and actual attitude angle；
Step 3：Sliding Mode Attitude design of control law：Quadrotor nonlinear attitude kinetic model is set up, chooses and slides
Dynamic model state and Reaching Law, design Sliding Mode Attitude control law, obtain attitude control quantity；
Step 4：Adaptive fuzzy sliding mode attitude control law is designed：Using sliding mode and its first differential as fuzzy control
The input of device, to control gain as the output design adaptive fuzzy sliding mode attitude control law of fuzzy controller, passes through fuzzy rule
Then on-line tuning control gain, to suppress buffeting caused by sliding formwork control.
Wherein, the instruction attitude angle described in step one is Ωr=[θr,ψr,φr]T, θr、ψr、φrRespectively instruction is bowed
The elevation angle, yaw angle and roll angle, subscript T represent the transposition of vector or matrix.
Wherein, the margin of error between the computations attitude angle and actual attitude angle described in step 2, its calculating side
Method is：
E=Ωr- Ω=[θr-θ,ψr-ψ,φr-φ]T (1)
Ω=[θ, ψ, φ]TFor actual attitude angle, θ, ψ, φ are respectively the actual angle of pitch, yaw angle and roll angle, are seen
Shown in Fig. 2.
Wherein, the design Sliding Mode Attitude control law described in step 3, its method is as follows：
1) quadrotor attitude dynamics model is set up
The attitude kinematics equations of quadrotor are：
In formula,For attitude angular rate, ω is attitude angular velocity, J (Ω) represent attitude angular rate and attitude angular velocity it
Between transition matrix, its expression formula is：
The attitude dynamic equations of quadrotor are：
In formula, τ=[L, M, N]TFor control moment, L, M and N are respectively rolling moment, pitching moment and yawing,
For the first differential of attitude angular velocity, I is inertia matrix, and its expression formula is：
In formula, Ix、IyAnd IzFor around three rotary inertias of body reference axis.
2) Sliding Mode Attitude control law is designed
The attitude dynamic equations that the attitude kinematics equations and formula (4) that formula (2) is represented are represented constitute four rotor flyings
The nonlinear attitude kinetic model of device, using the nonlinear attitude kinetic model as controlled device, design Sliding Mode Attitude control
Rule, obtains controlled quentity controlled variable, its design method is：
1. sliding mode is chosen
Choose following sliding mode：
In formula, s is sliding mode,For margin of error e first differential, c=diag (c1,c2,c3), diag () expressions pair
Angular moment battle array, c1、c2、c3It is arithmetic number.
2. exponentially approaching rule is chosen：
Choose following exponentially approaching rule：
In formula,For sliding mode s first differential, λ=diag (λ1,λ2,λ3),λ1、λ2、λ3It is arithmetic number, k=
diag(k1,k2,k3), k1、k2、k3It is arithmetic number, diag () represents diagonal matrix, and sign (s) represents s sign function.
3. Sliding Mode Attitude control law is designed
Design following Sliding Mode Attitude control law：
In formula, J-1(Ω) represents J (Ω) inverse matrix,For margin of error e first differential,Represent J (Ω) single order
Differential,Represent ΩrSecond-order differential, sign (s) represent s sign function.
Include sign function ksign (s), therefore, attitude control law in Sliding Mode Attitude control law represented by formula (8)
Being toggled between different control logics causes to buffet, so as to influence the dynamic property of control system.For this problem, sheet
Invention devises adaptive fuzzy sliding mode attitude control law, suppresses to buffet with effective.
Wherein, the design adaptive fuzzy sliding mode attitude control law described in step 4, with sliding mode and its single order
Differential is the input of fuzzy controller, to control gain as the output design adaptive fuzzy sliding mode gesture stability of fuzzy controller
Rule, gain is controlled by fuzzy rule on-line tuning, and to suppress to buffet, its method is as follows：
1. input/output variable is selected
The input for making fuzzy controller is sliding mode s=[s1,s2,s3]TAnd its first differentialWherein,
s1,s2,s3The sliding mode that respectively pitching, driftage and roll channel are taken,Respectively s1,s2,s3First differential,
Output variable is k=diag (k1,k2,k3), it is possible thereby to according to s andChange on-line tuning control gain k value.
2. the fuzzy set of input/output variable is defined
The fuzzy subset of description input variable and output variable may be defined as：{ NB, NS, ZO, PS, PB }, wherein, NB is negative
Greatly, NS be bear small, ZO be zero, PS be just small, PB is honest.
3. fuzzy rule is determined
Using IF-THEN (if-so) fuzzy rule：
R(j):If siForAndForThen kiFor Bj。
Wherein, R(j)For fuzzy rule sentence, subscript j represents j-th strip fuzzy rule,For variable siThe son of fuzzy set
Collection,For variableThe subset of fuzzy set, siFor vectorial s i-th of element, kiFor vectorial k i-th of element, i value
For 1,2,3,For vectorI-th of element, i value is 1,2,3, BjFor the output of j-th strip fuzzy rule.
4. de-fuzzy
Using product inference machine, monodrome fuzzy device and center method of average de-fuzzy, output variable is obtained：
In formula, siSubscript i value can be 1,2,3,Represent siFuzzy membership function, l and n represent fuzzy
Regular number,
Represent fuzzy membership functionIntermediate value,
Thus, by the algorithm described in formula (9) can on-line tuning control gain k, to suppress sign function ksign
(s) buffeted caused by.
A kind of quadrotor attitude control method of the present invention, compared with prior art, its advantage is：
1) attitude control method proposed by the present invention, using nonlinear attitude kinetic model as controlled device, it is contemplated that non-
Coupling between linear term and passage, improves the adaptability of system.
2) this method devises Sliding Mode Attitude control law so that system by choosing suitable sliding mode and Reaching Law
It is uncertain to model and external disturbance that there is good robustness.
3) this method is using sliding mode and its first differential as the input of fuzzy controller, to control gain as fuzzy control
The output design adaptive fuzzy sliding mode attitude control law of device, controls gain by fuzzy rule on-line tuning, can effectively press down
Buffeted caused by sliding formwork control processed, improve the dynamic property of system.
Engineer is controlled to give arbitrary instruction attitude angle according to actual quadrotor in application process, and will
The controlled quentity controlled variable obtained by this method, which is transmitted to executing agency, realizes gesture stability.
Brief description of the drawings
Fig. 1 is control system architecture figure of the present invention；
Fig. 2 is quadrotor attitude motion schematic diagram of the present invention；
Fig. 3 is Sliding Mode Attitude control result figure of the present invention；
Fig. 4 is Sliding Mode Attitude control input figure of the present invention；
Fig. 5 is adaptive fuzzy sliding mode gesture stability result figure of the present invention；
Fig. 6 is that adaptive fuzzy sliding mode gesture stability of the present invention inputs figure.
Symbol description is as follows in figure：
Ω Ω=[θ, ψ, φ]TFor the actual attitude angle of quadrotor, wherein θ, ψ, φ is respectively actual pitching
Angle, yaw angle and roll angle；
ΩrΩr=[θr,ψr,φr]TFor the instruction attitude angle of quadrotor, wherein θr、ψr、φrRespectively instruction is bowed
The elevation angle, instruction yaw angle and instruction roll angle；
C c=diag (c1,c2,c3) be sliding mode design parameter；
E e=Ωr- Ω is the margin of error between instruction attitude angle and actual attitude angle；
Instruct the first differential of the margin of error between attitude angle and actual attitude angle；
ω attitude angular velocities；
S sliding modes；
The first differential of sliding mode；
K k=diag (k1,k2,k3) it is control gain；
Du/dt differentiates；
Σ summation operations；
τ τ=[L, M, N]TFor control input；
L rolling moments；
M pitching moments；
N yawings；
oexeyezeEarth axes；
Oxyz body coordinate systems；
P angular velocity in roll；
Q rate of pitch；
R yaw rates；
Embodiment
A kind of quadrotor attitude control method, it is comprised the following steps that：
Step one：Given instruction attitude angle
Given instruction attitude angle is Ωr=[0.1rad, 0.2rad, -0.05rad]T, θr、ψr、φrRespectively instruct pitching
Angle, yaw angle and roll angle.
Step 2：The margin of error is calculated
The margin of error between computations attitude angle and actual attitude angle：E=Ωr- Ω=[θr-θ,ψr-ψ,φr-φ]T,
Wherein, Ω=[θ, ψ, φ]TFor actual attitude angle, θ, ψ, φ are respectively the actual angle of pitch, yaw angle and roll angle, are continuous
Changing value.
Step 3：
1) quadrotor attitude dynamics model is set up
The attitude kinematics equations of quadrotor are：
In formula,For attitude angular rate, ω is attitude angular velocity, J (Ω) represent attitude angular rate and attitude angular velocity it
Between transition matrix, its expression formula is：
The attitude dynamic equations of quadrotor are：
In formula, τ=[L, M, N]TFor control moment, L, M and N are respectively rolling moment, pitching moment and yawing,
For the first differential of attitude angular velocity, I is inertia matrix, and its expression formula is：
In formula, Ix、IyAnd IzFor around three rotary inertias of body reference axis.
The major parameter of quadrotor is as shown in table 1.
The major parameter of the quadrotor of table 1
2) Sliding Mode Attitude control law is designed
Using nonlinear attitude kinetic model as controlled device, Sliding Mode Attitude control law is designed, system control amount is obtained, its
Design method is：
1. sliding mode is chosen
Choose following sliding mode：
In formula, c=diag (2,2,2).
2. exponentially approaching rule is chosen：
Choose following exponentially approaching rule：
In formula, λ=diag (0.1,0.1,0.1), k=diag (0.3,0.3,0.3).
3. Sliding Mode Attitude control law is designed
Design following Sliding Mode Attitude control law：
In formula, J-1(Ω) represents J (Ω) inverse matrix,Represent ΩrSecond-order differential,Single order for margin of error e is micro-
Point,J (Ω) first differential is represented, sign (s) represents s sign function.
Step 4：Design adaptive fuzzy sliding mode attitude control law：
Using sliding mode and its first differential as the input of fuzzy controller, to control gain as the output of fuzzy controller
Adaptive fuzzy sliding mode attitude control law is designed, gain is controlled by fuzzy rule on-line tuning, to suppress to buffet, its design side
Method is as follows：
1. input/output variable is selected
The input for making fuzzy controller is sliding mode s=[s1,s2,s3]TAnd its first differentialOutput
Variable is k=[k1,k2,k3]T。
2. the fuzzy set of input/output variable is defined
The fuzzy subset of description input variable and output variable may be defined as：{ NB, NS, ZO, PS, PB }, wherein, NB is negative
Greatly, NS be bear small, ZO be zero, PS be just small, PB is honest.
3. fuzzy rule is determined
Using IF-THEN fuzzy rules：
R(j):If siForAndForThen kiFor Bj。
Wherein, R(j)For fuzzy rule sentence, subscript j represents j-th strip fuzzy rule,For variable siThe son of fuzzy set
Collection,For variableThe subset of fuzzy set, kiFor vectorial k i-th of element, siFor vectorial s i-th of element,For vector
I-th of element, i value is 1,2,3, BjFor the output of j-th strip fuzzy rule.The fuzzy rule that the present invention is set such as following table
Shown in 2.
The fuzzy reasoning table of table 2
4. de-fuzzy
Using product inference machine, monodrome fuzzy device and center method of average de-fuzzy, output variable is obtained：
In formula, siSubscript i value can be 1,2,3,Represent siFuzzy membership function, l and n represent fuzzy
Regular number,
Represent fuzzy membership functionIntermediate value,
Membership function is chosen as follows：
Thus, by above-mentioned adaptive fuzzy sliding mode gesture stability can on-line tuning control gain k, to suppress symbol
Buffeted caused by function item k sign (s).
Quadrotor gesture stability result in embodiment is as shown in figures 3 to 6.Fig. 3 gives Sliding Mode Attitude control
As a result, Fig. 4 gives Sliding Mode Attitude control input, and Fig. 5 gives adaptive fuzzy sliding mode gesture stability result, and Fig. 6 gives
Adaptive fuzzy sliding mode gesture stability is inputted.It can be obtained by Fig. 3 and Fig. 5：Attitude control method proposed by the invention can make four
The attitude angle of rotor craft is adjusted to instruction attitude angle exactly, illustrates the validity of this method；In Fig. 4 and Fig. 6
Control input contrast can be obtained：Adaptive fuzzy sliding mode gesture stability can effectively weaken buffeting.
Claims (1)
1. a kind of quadrotor attitude control method, it is characterised in that step is as follows：
Step one：Given instruction attitude angle, gives instruction pitching angle thetar, instruct yaw angle ψr, instruction roll angle φr；
Instruction attitude angle is Ωr=[θr,ψr,φr]T, θr、ψr、φrThe respectively instruction angle of pitch, instruction yaw angle and instruction rolling
Angle, subscript T represents the transposition of vector or matrix；
Step 2：The margin of error is calculated, the margin of error e between computations attitude angle and actual attitude angle；
E=Ωr- Ω=[θr-θ,ψr-ψ,φr-φ]T (1)
Ω=[θ, ψ, φ]TFor actual attitude angle, θ, ψ, φ are respectively the actual angle of pitch, yaw angle and roll angle；
Step 3：Sliding Mode Attitude design of control law：Quadrotor nonlinear attitude kinetic model is set up, sliding die is chosen
State and Reaching Law, design Sliding Mode Attitude control law, obtain attitude control quantity；
1) quadrotor attitude dynamics model is set up
The attitude kinematics equations of quadrotor are：
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In formula,For attitude angular rate, ω is attitude angular velocity, and J (Ω) represents turning between attitude angular rate and attitude angular velocity
Matrix is changed, its expression formula is：
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<mo>-</mo>
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<mo>(</mo>
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The attitude dynamic equations of quadrotor are：
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<mi>I</mi>
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<mi>&omega;</mi>
<mo>&CenterDot;</mo>
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<mo>+</mo>
<mi>&omega;</mi>
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<mo>-</mo>
<mo>-</mo>
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<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, τ=[L, M, N]TFor control moment, L, M and N are respectively rolling moment, pitching moment and yawing,For appearance
The first differential of state angular speed, I is inertia matrix, and its expression formula is：
<mrow>
<mi>I</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
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<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
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<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, Ix、IyAnd IzFor around three rotary inertias of body reference axis；
2) Sliding Mode Attitude control law is designed
The attitude dynamic equations that the attitude kinematics equations and formula (4) that formula (2) is represented are represented constitute quadrotor
Nonlinear attitude kinetic model, using the nonlinear attitude kinetic model as controlled device, designs Sliding Mode Attitude control law, obtains
To controlled quentity controlled variable, its design method is：
1. sliding mode is chosen
Choose following sliding mode：
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<mo>=</mo>
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In formula, s is sliding mode,For margin of error e first differential, c=diag (c1,c2,c3), diag () is represented to angular moment
Battle array, c1、c2、c3It is arithmetic number；
2. exponentially approaching rule is chosen：
Choose following exponentially approaching rule：
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<mi>S</mi>
<mo>&CenterDot;</mo>
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In formula,For sliding mode s first differential, λ=diag (λ1,λ2,λ3),λ1、λ2、λ3It is arithmetic number, k=diag (k1,
k2,k3), k1、k2、k3It is arithmetic number, diag () represents diagonal matrix, and sign (s) represents s sign function；
3. Sliding Mode Attitude control law is designed
Design following Sliding Mode Attitude control law：
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In formula, J-1(Ω) represents J (Ω) inverse matrix,For margin of error e first differential,Represent that J (Ω) single order is micro-
Point,Represent ΩrSecond-order differential, sign (s) represent s sign function；
Step 4：Adaptive fuzzy sliding mode attitude control law is designed：Using sliding mode and its first differential as fuzzy controller
Input, to control gain as the output design adaptive fuzzy sliding mode attitude control law of fuzzy controller, is existed by fuzzy rule
Line adjustment control gain, to suppress buffeting caused by sliding formwork control；
1. input/output variable is selected
The input for making fuzzy controller is sliding mode s=[s1,s2,s3]TAnd its first differentialWherein, s1,
s2,s3The sliding mode that respectively pitching, driftage and roll channel are taken,Respectively s1,s2,s3First differential, it is defeated
Go out variable for k=diag (k1,k2,k3), it is possible thereby to according to s andChange on-line tuning control gain k value；
2. the fuzzy set of input/output variable is defined
The fuzzy subset of description input variable and output variable may be defined as：{ NB, NS, ZO, PS, PB }, wherein, NB for it is negative big,
NS be bear small, ZO be zero, PS be just small, PB is honest；
3. fuzzy rule is determined
Using IF-THEN fuzzy rules：
R(j):If siForAndForThen kiFor Bj；
Wherein, R(j)For fuzzy rule sentence, subscript j represents j-th strip fuzzy rule,For variable siThe subset of fuzzy set,
For variableThe subset of fuzzy set, kiFor vectorial k i-th of element, siFor vectorial s i-th of element,For vectorI-th
Individual element, i value is 1,2,3, BjFor the output of j-th strip fuzzy rule；
4. de-fuzzy
Using product inference machine, monodrome fuzzy device and center method of average de-fuzzy, output variable is obtained：
<mrow>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>l</mi>
</munderover>
<msubsup>
<mi>&gamma;</mi>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mi>j</mi>
</msubsup>
<mrow>
<mo>(</mo>
<munderover>
<mo>&Pi;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</munderover>
<msub>
<mi>&mu;</mi>
<msubsup>
<mi>F</mi>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mi>j</mi>
</msubsup>
</msub>
<mo>(</mo>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>l</mi>
</munderover>
<mrow>
<mo>(</mo>
<munderover>
<mo>&Pi;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</munderover>
<msub>
<mi>&mu;</mi>
<msubsup>
<mi>F</mi>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mi>j</mi>
</msubsup>
</msub>
<mo>(</mo>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<msubsup>
<mi>&Gamma;</mi>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mi>T</mi>
</msubsup>
<msub>
<mi>&xi;</mi>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, siSubscript i value can be 1,2,3,Represent siFuzzy membership function, l and n represent fuzzy rule
Number,
Represent fuzzy membership functionIntermediate value,
<mrow>
<msub>
<mi>&xi;</mi>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<msubsup>
<mi>&xi;</mi>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mn>1</mn>
</msubsup>
<mrow>
<mo>(</mo>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<msubsup>
<mi>&xi;</mi>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mi>j</mi>
</msubsup>
<mrow>
<mo>(</mo>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<msubsup>
<mi>&xi;</mi>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mi>i</mi>
</msubsup>
<mrow>
<mo>(</mo>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>,</mo>
<msubsup>
<mi>&xi;</mi>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mi>j</mi>
</msubsup>
<mrow>
<mo>(</mo>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<munderover>
<mo>&Pi;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</munderover>
<msub>
<mi>&mu;</mi>
<msubsup>
<mi>F</mi>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mi>j</mi>
</msubsup>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>l</mi>
</munderover>
<mrow>
<mo>(</mo>
<munderover>
<mo>&Pi;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</munderover>
<msub>
<mi>&mu;</mi>
<msubsup>
<mi>F</mi>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mi>j</mi>
</msubsup>
</msub>
<mo>(</mo>
<msub>
<mi>s</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>;</mo>
</mrow>
Thus, by the algorithm described in formula (9) can on-line tuning control gain k, led with suppressing sign function ksign (s)
The buffeting of cause.
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| CN107562064B (en) * | 2016-06-30 | 2020-08-25 | 北京电子工程总体研究所 | Attitude control distribution method of aircraft based on multiple actuating mechanisms |
| CN106708082B (en) * | 2017-03-21 | 2019-06-21 | 烟台南山学院 | Aircraft pitch channel attitude command fast tracking method based on fuzzy control |
| CN106814626B (en) * | 2017-03-21 | 2019-06-14 | 烟台南山学院 | Supersonic aircraft overload instruction fast tracking method based on fuzzy variable structure |
| CN108153148B (en) * | 2017-07-03 | 2020-02-21 | 浙江工业大学 | Enhanced index approach law sliding mode control method of quad-rotor unmanned aerial vehicle system |
| CN107368089B (en) * | 2017-07-11 | 2019-12-03 | 浙江工业大学 | A kind of quadrotor nonlinear sliding mode posture control method based on double exponential type functions |
| CN107561931B (en) * | 2017-07-11 | 2019-12-03 | 浙江工业大学 | A kind of quadrotor nonlinear sliding mode posture control method based on monoexponential function |
| CN107368088B (en) * | 2017-07-11 | 2019-12-03 | 浙江工业大学 | A kind of quadrotor nonlinear sliding mode posture control method based on error extension type function |
| CN107491083B (en) * | 2017-09-11 | 2020-10-09 | 北京航空航天大学 | Four-rotor-wing autonomous landing method based on saturation self-adaptive sliding mode control |
| CN108549208B (en) * | 2018-03-14 | 2021-12-17 | 重庆邮电大学 | Four-rotor aircraft attitude control method based on factor self-adaptive fuzzy PID |
| CN108549225B (en) * | 2018-04-12 | 2020-02-21 | 浙江工业大学 | Rigid aerospace vehicle finite time self-adaptive fault-tolerant control method based on enhanced power-order approach law and fast terminal sliding mode surface |
| CN108549401B (en) * | 2018-05-28 | 2021-02-26 | 浙江工业大学 | Finite time control method of four-rotor aircraft based on hyperbolic sine enhanced index approach law and fast terminal sliding mode surface |
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